Revision as of 11:05, 10 January 2010 by imported>Peter Schmitt
A geometric series is a series associated with a geometric sequence,
i.e., the ratio (or quotient) q of two consecutive terms is the same for each pair.
An infinite geometric series converges if and only if |q|<1.
Then its sum is where a is the first term of the series.
Examples
Positive ratio
The series
and corresponding sequence of partial sums
is a geometric series with quotient
and first term
and therefore its sum is
Negative ratio
The series
and corresponding sequence of partial sums
is a geometric series with quotient
and first term
and therefore its sum is
Power series
Any geometric series
can be written as
where
The partial sums of the power series Σxk are
because
Since
it is
and the geometric series converges (more precisely: converges absolutely) for |x|<1 with the sum
and diverges for |x| ≥ 1.
(Depending on the sign of a, the limit is +∞ or −∞ for x≥1.)